A Mathematical Exam About Ellipse & Hyperbola. Be A Mathematician

1. The standard equation 9x² - 4y² = 36 is

Option 1

Option 2

Option 3

Option 4

The standard equation 9x^2 - 4y^2 = 36 represents a hyperbola. The coefficients of x^2 and y^2 are positive and different, indicating that the hyperbola opens horizontally. The equation is in the standard form of a hyperbola, where the squares of x and y are divided by positive constants and subtracted to equal a constant. Therefore, the correct answer is Option 1.

Explanation

The standard equation 9x^2 - 4y^2 = 36 represents a hyperbola. The coefficients of x^2 and y^2 are positive and different, indicating that the hyperbola opens horizontally. The equation is in the standard form of a hyperbola, where the squares of x and y are divided by positive constants and subtracted to equal a constant. Therefore, the correct answer is Option 1.

2. What

Option 1

Option 2

Option 3

Option 4

not-available-via-ai

Explanation

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3. The segment made by the endpoints of the vertices of a hyperbola is called

Conjugate axis

Major axis

Minor axis

Transverse axis

The segment made by the endpoints of the vertices of a hyperbola is called the transverse axis. This axis is the line segment that passes through the center of the hyperbola and is perpendicular to the conjugate axis. It is the longest segment in the hyperbola and determines the overall shape and orientation of the hyperbola.

Explanation

The segment made by the endpoints of the vertices of a hyperbola is called the transverse axis. This axis is the line segment that passes through the center of the hyperbola and is perpendicular to the conjugate axis. It is the longest segment in the hyperbola and determines the overall shape and orientation of the hyperbola.

4. Is [e = .98 an ellipse] true or false?

True

False

The statement "e = .98 an ellipse" is true. In mathematics, e represents the eccentricity of an ellipse. An ellipse is a closed curve that resembles a flattened circle, and its eccentricity value must be between 0 and 1. Therefore, if e = .98, it falls within the valid range and satisfies the definition of an ellipse.

Explanation

The statement "e = .98 an ellipse" is true. In mathematics, e represents the eccentricity of an ellipse. An ellipse is a closed curve that resembles a flattened circle, and its eccentricity value must be between 0 and 1. Therefore, if e = .98, it falls within the valid range and satisfies the definition of an ellipse.

5. Is [Ellipse the best structure to follow in making a Cooling Tower in powerplant because of its efficiency in heat removing process ?] true or false?

True

False

The statement suggests that an ellipse is the best structure to follow in making a cooling tower in a power plant because of its efficiency in the heat removing process. However, this statement is false. The efficiency of a cooling tower in the heat removing process is not determined by its shape, such as an ellipse. The efficiency of a cooling tower depends on various factors such as the design, size, materials used, and the specific cooling requirements of the power plant. Therefore, it is incorrect to claim that an ellipse is the best structure for a cooling tower in a power plant.

Explanation

The statement suggests that an ellipse is the best structure to follow in making a cooling tower in a power plant because of its efficiency in the heat removing process. However, this statement is false. The efficiency of a cooling tower in the heat removing process is not determined by its shape, such as an ellipse. The efficiency of a cooling tower depends on various factors such as the design, size, materials used, and the specific cooling requirements of the power plant. Therefore, it is incorrect to claim that an ellipse is the best structure for a cooling tower in a power plant.

6. It is the set of points in plane such that the difference of its distances from two fixed points is constant

Circle

Parabola

Ellipse

Hyperbola

A hyperbola is defined as the set of points in a plane such that the difference of its distances from two fixed points (called the foci) is constant. This means that for any point on the hyperbola, the difference between its distance to one focus and its distance to the other focus is always the same. This property distinguishes a hyperbola from other conic sections like a circle, parabola, or ellipse, which have different distance properties. Therefore, the given description aligns with the definition of a hyperbola.

Explanation

A hyperbola is defined as the set of points in a plane such that the difference of its distances from two fixed points (called the foci) is constant. This means that for any point on the hyperbola, the difference between its distance to one focus and its distance to the other focus is always the same. This property distinguishes a hyperbola from other conic sections like a circle, parabola, or ellipse, which have different distance properties. Therefore, the given description aligns with the definition of a hyperbola.

7. A segment passing through the focus of the ellipse/hyperbola that is perpendicular to the major axis.

Asymptote

Directix

Latus rectum

Transverse axis

The latus rectum is a segment passing through the focus of the ellipse/hyperbola that is perpendicular to the major axis. It is a key property of conic sections and is used to define the shape and dimensions of the ellipse or hyperbola. The latus rectum plays an important role in determining the focal length and eccentricity of the conic section.

Explanation

The latus rectum is a segment passing through the focus of the ellipse/hyperbola that is perpendicular to the major axis. It is a key property of conic sections and is used to define the shape and dimensions of the ellipse or hyperbola. The latus rectum plays an important role in determining the focal length and eccentricity of the conic section.

8. What is area of an ellipse?

The area of an ellipse is given by the formula πAB, where A and B are the lengths of the semi-major and semi-minor axes of the ellipse, respectively. This formula is derived from the mathematical definition of an ellipse and is used to calculate its area.

Explanation

The area of an ellipse is given by the formula πAB, where A and B are the lengths of the semi-major and semi-minor axes of the ellipse, respectively. This formula is derived from the mathematical definition of an ellipse and is used to calculate its area.

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